where \(\rho >0\) , \(c(\Omega )>0\)  is a constant the depends only on the domain \(\Omega \) , and \({\rm Surf}\)  is a surface measure, e.g., the co-dimension 1 Hausdorff measure. The first inequality of this type was proven by Steinerberger [1]} for two-dimensional domains such as the unit square \([0,1]^2\) , with \(\rho =1\) . This result was then generalized to certain compact domains in \(\mathbb {R}^d\)  with arbitrary \(d\ge 1\)  by Steinerberger and the second author with \(\rho = 4-1/d\) , see [2]}.
Subsequently, Carroll, Massaneda, and Ortega-Cerdà [3]} improved the exponent to \(\rho = 2-1/d\) , and Cavalletti and Farinelli proved that the optimal \(\rho =1\)  is true in all dimensions [4]}. The analysis in [4]} takes place at a much larger class of metric spaces that satisfy the Curvature-Dimension (\({\rm CD}\) ) condition, where the nodal set can be measured using the notion of Perimeter [6]}, [7]}.
